The Complexity of Sampling Histories

Scott Aaronson, UC Berkeley

http://www.cs.berkeley.edu/~aaronson

August 5, 2003

Words You Should Stop Me If I Use

polysize

oracle

relativizing

zero-knowledge

#P-complete

nonuniform

Words I’ll Stop You If

You Use

holonomy

gauge

SU(2)

intertwinor

kinematical

Lagrangian

Words To Be Careful

With

loop

string

Outline

• Why you should stay up at night worrying about quantum mechanics

• Dynamical quantum theories

• Solving Graph Isomorphism by sampling histories

• Search in N1/3 queries (but not fewer)

What weexperience

Quantum mechanics

Assumption

1 1 1 11 2 3 41 2 3 4

Time

2 2 2 21 2 3 41 2 3 4

3 3 3 31 2 3 41 2 3 4

4 4 4 41 2 3 41 2 3 4

Quantum state of the universe

You

A Puzzle• Let |OR = you seeing a red dot

|OB = you seeing a blue dot

2

( )

1

:

:

R R B B

H

R R B B

t O O

t O O

• What is the probability that you see the dot change color?

The Goal

1 11 1 1

1

N

N N NN N

u u

u u

2 2

1 11 1 1

2 21

N

N NNN N

s s

s s

Quantum state

Quantum state

Probability distribution

Probability distribution

Unitary matrix

Stochastic matrix

Why Look for This?

• Quantum theory says nothing about multiple-time or transition probabilities

• Then what is a “prediction,” or the “output of a computation,” or the “utility of a decision”?

• Reply:

“But we have no direct knowledge of the past anyway, just records”

Bohm’s Theory

• Gives a deterministic evolution rule for particle positions and momenta

• Mathematicianly approach: Study the set of all discrete dynamical rules, without presupposing one of them is “true”

• But doesn’t make sense for discrete observables: 1 1 1

12 2 21 1 1 0

2 2 2

Our Results

• We give evidence that by examining a history, one could solve problems that are intractable even for a quantum computer

- Graph Isomorphism and Approximate Shortest Lattice Vector in polynomial time

- Unordered search in N1/3 steps instead of N1/2

• We define dynamical theories for obtaining classical histories, and investigate what axioms they can satisfy

• We obtain the first model of computation “slightly” more powerful than quantum computing

Dynamical Theory

• Must marginalize to single-time probabilities of quantum mechanics: diagonal entries of and UU-1

• Fix an N-dimensional Hilbert space (N finite) and orthogonal basis

• Given an NN unitary U and state acted on, returns a stochastic matrix UDS ,

Axiom: Symmetry

D is invariant under relabeling of basis states:

111 ,, PUQDQUPPPD

Axiom: Indifference

If U acts on and is the identity on H2, then S should also be the identity on H2

Can formalize without tensor products: partition U into minimal blocks of nonzero entries

Not the same as commutativity:

1 1, , , ,A AB A B AB A B AB B A AB BD U U U D U D U U U D U

21 HH

Theorem: No dynamical theory satisfies both indifference and commutativity

Proof: Suppose A and B share an EPR pair

UA applies /8 rotation to first qubit, UB applies -/8 to second qubit. Consider probability p of being at |00 initially and |10 at the end

.2/1100

2 2

2 2

1 1cos 00 sin 01

2 8 2 8

1 1sin 10 cos 11

2 8 2 8

If UA applied first:

8sin

2

1 2 p

2 2

2 2

1 1cos 00 sin 01

2 8 2 8

1 1sin 10 cos 11

2 8 2 8

If UB applied first:

2 21 1 1sin sin

4 2 8 2 8p

Axiom: Robustness

Small (1/poly(N)) change to or U

Small (1/poly(N)) change to jointprobabilities matrix, S·diag()

Arguably that’s needed for any physical theory or model of computation

Example 1: Product Dynamics

Symmetric, robust, commutative, but not indifferent

2 2 2 2

1 1 1 1

2 2 2 2

N N N N

Take probabilities at any two times to be independent of each other

Example 2: Dieks Dynamics

Symmetric, indifferent, but not commutative or robust

4 / 5 0 1 3/ 5

3/ 5 1 0 4 / 5

2 2

2 2

4 / 5 3/ 50 1

1 03/ 5 4 / 5

Partition U into minimal blocks, then apply product dynamics separately to each

Theorem: Suppose

Then there is a weight-1 “flow” through the network

where flow through an edge can’t exceed the edge’s capacity

.1

1

1111

NNNN

N

N uu

uu

NNu

2

1

2

N2

N

2

1

st

11u

Proof Idea: By the Max-Flow-Min-Cut Theorem (Ford-Fulkerson 1956), it suffices to show that any set of edges separating s from t (a cut) has total capacity at least 1. Let A,B be right, left edges respectively not in cut C. Then the capacity of C is

so we need to show

Fix U and consider maximum of right-hand side. Equals the max eigenvalue of a positive semidefinite matrix, which we can analyze using some linear algebra…

22

,i j ij

i A j B i A j B

u

.122

,

Bj

jAi

iBjAiiju

Example 3: Flow Dynamics

Using the previous theorem, we construct a dynamical theory that satisfies the symmetry, indifference, and robustness axioms

Not obvious a priori that any such theory exists

Model of Computation

• Oracle chooses a symmetric robust indifferent theory D “adversarially,” then returns a sample from D

• Polynomial-time classical computation, with one query to a history oracle

• Oracle takes as input descriptions of quantum circuits U1,…,UT

• Any dynamical theory D induces a distribution D over classical histories for

1 10 0 0n n n

TU U U

• At least as powerful as standard quantum computing

The Graph Isomorphism Problem• Decide whether two graphs G and H are isomorphic

• The best known algorithm takes about timen = number of vertices

• But we don’t think Graph Isomorphism is NP-complete

• Intuitively, it’s “only” as hard as counting collisions in

Could be easier than finding a needle in a haystack!

1 ! 1 !, , , , ,n nG G H H

2 n

The Collision Problem

• Given a list of N numbers x1,…,xN, you’re promised that either every number occurs once, or every number occurs twice. Decide which.

• Best classical algorithm makes ~ queries (“birthday paradox”)

• Brassard, Høyer, Tapp 1997 gave a quantum algorithm that makes ~N1/3 queries

• Is there a faster quantum algorithm—say, log N queries? If so, we’d get a polynomial-time quantum algorithm for Graph Isomorphism!

3 6 1 5 4 2 vs. 6 2 2 5 6 5

N

The Collision Problem (con’t)• Aaronson 2002: Any quantum algorithm needs at least ~N1/5 queries

• Improved by Shi to ~N1/3 queries

• Previously, couldn’t even rule out constant number of queries!

• Proofs use multivariate polynomials

• Implications:

• No “dumb” quantum algorithm for Graph Isomorphism

• “Oracle separation” between the complexity classes BQP (Bounded-Error Quantum Polynomial-Time) and DQP (Dynamical Quantum Polynomial-Time)

PPolynomial

Time

BQPQuantum

PolynomialTime

DQPMy New Class

NPSatisfiability, Traveling

Salesman, etc.

Factoring

Graph Isomorphism

Approximate Shortest Vector

ConjecturedWorld Map

Solving the Collision Problem by Sampling Histories

1

1 1

2

N

i ii

i x i j xN

GOAL: When we inspect the classical history, see both |i and |j with high probability

Suppose every number occurs twice. Then

“Measurement” of 2nd register

Two bitwise Fourier transforms

1

2ii j x

Solving the Collision Problem by Sampling Histories (con’t)

Theorem: Under any dynamical theory satisfying the symmetry and indifference axioms, the first Fourier transform makes the hidden variable “forget” whether it was at |i or |j. So after the second Fourier transform, it goes to |i half the time and |j half the time; thus with ½ probability we see both |i and |j in the history

Proof Idea: Use symmetry axiom, together with automorphisms of

Indifference axiom needed to “trace out” second register

2nZ

Finding a Marked Item in N1/3 Queries

N1/3 iterations

of Grover’s quantum search

algorithm

Probability of observing the

marked item after T iterations is ~T2/N

Hidden variable

N1/3 Search Algorithm Is Optimal

• Bennett, Bernstein, Brassard, Vazirani 1996: If a quantum computer searches a list of N items for a single randomly-placed marked item, the probability of observing the marked item after T steps is at most

• So probability of observing it in a history of the first T steps is at most

2T

N

2 3

1

T

t

t T

N N

• Summary: If “your whole life flashed before you in an instant,” and if you’d prepared for this by putting your brain in certain superpositions, then (under reasonable axioms) you could solve Graph Isomorphism in polynomial time

• Contrast: Nonlinear quantum mechanics would put Satisfiability and even harder problems in polynomial time (Abrams and Lloyd 1998)

• But probably still not Satisfiability

• Postulate: NP-complete problems can’t be efficiently solved in physical reality

• The postulate does not imply your whole life couldn’t flash before you in an instant

• Justification for the postulate: Maybe I’m wrong, but then I’d be too busy solving NP-complete problems to care that I was wrong

(1) Quantum states evolve linearly

(2) We can’t make unlimited-precision measurements

(3) The “self-sampling” anthropic principle (Bostrom 2000) is false(4) Constraints on quantum gravity?

• What does the postulate imply? (under plausible complexity assumptions)